We apply the principal component inverse (PCI) method of rapidly adaptive interference suppression to the problem of separating the strong components of reverberation in an active reverberation experiment. A best low-rank estimate of a reverberation return is extracted by decomposing the returns into short duration subintervals over which the signal is approximately sinusoidal and by invoking the Eckart-Young theorem. This establishes the connection between the best approximation of a data matrix and its singular value decomposition (SVD). The signal rank used in the approximation is adaptively determined using a background noise estimate as a threshold on the sums of squares of the singular values. We show that for known transmit waveforms, in particular a hyperbolic frequency-modulated (HFM) sinusoid, performance of the standard PCI approach may be improved by appropriately transforming the data prior to the SVD. This transformation reduces the degree of subspace smearing thus allowing a cleaner separation of the signal plus noise and noise subspaces. HFM reverberation data from the Acoustic Reconnaissance Cruise of the Acoustic Reverberation Special Research Program (ARSRP) corresponding to known bottom features are analyzed using this PCI method. A physical interpretation of the low-rank PCI reverberation estimate is developed by comparing the PCI estimate to estimates derived from a parametric approach. We show that the PCI estimate retains the salient reverberation signal characteristics required for subsequent parametric modeling. The parametric estimates are performed using the fast maximum likelihood (FML) algorithm which can efficiently and effectively carry out the least-squares fitting of a complicated signal model, with many components and nonlinearly entering parameters. The PCI technique is also applied to the problem of detecting a weak signal masked by the stronger reverberation returns. In this application the detection processing is performed on the PCI residual obtained by removing the strong reverberation returns.